Animated Solution for Physics - Gravitation: Consider a planet in some solar system which has a mass double the mass of Earth and density equal to the average density of Earth. If the weight of an object on Earth is $w$, the weight of the same object on that planet will be

Imagine you are an interstellar traveler, stepping off your spaceship onto a newly discovered planet. You know this planet is twice as massive as Earth, but its average density is exactly the same as our home planet. The burning question is: how heavy will you feel? Will you be crushed by double the gravity, or is there a catch?

Let's break down the physics of weight and gravity to find out.

First, we need to understand what weight actually is. Weight ($w$) is the gravitational force a planet exerts on an object of mass $m$. Mathematically, it is expressed as:

Here, $g$ is the acceleration due to gravity on the surface of the planet. According to Newton's Law of Universal Gravitation, $g$ depends on the mass of the planet ($M$) and its radius ($R$):

If we just look at this equation, we might be tempted to say, "Hey, the mass $M$ is doubled, so $g$ must be doubled!" But that is a dangerous trap. We cannot assume the radius $R$ remains the same. A more massive planet with the same density must be physically larger.

To see how the radius changes, we need to express the planet's mass in terms of its density ($\rho$). Assuming the planet is a perfect sphere, its volume is $\frac{4}{3}\pi R^3$. Therefore, its mass is:

Now, let's substitute this expression for mass back into our equation for gravity:

Notice the beautiful cancellation that happens here. The $R^2$ in the denominator cancels out two powers of $R$ in the numerator, leaving us with a much simpler, elegant formula:

Look closely at this new equation for $g$. The values $\frac{4}{3}$, $\pi$, and $G$ are universal constants. Furthermore, the problem states that the density $\rho$ is the same for both Earth and the new planet.

This means that for these two planets, the acceleration due to gravity is directly proportional to their radius:

Consequently, the ratio of the weights on the two planets is simply the ratio of their radii:

We are almost there! We just need to find the ratio of their radii. Let's go back to our mass-density relationship:

Since density is constant, the mass is directly proportional to the cube of the radius ($M\propto R^3$). We can flip this around to say that the radius is proportional to the cube root of the mass:

We are given that the new planet is twice as massive as Earth ($M_p=2M_e$). Therefore, the ratio of their radii is:

Now, we substitute this radius ratio back into our weight ratio equation:

So, if your weight on Earth is $w$, your weight on this new, twice-as-massive planet will be $2^{1/3}w$. You won't feel twice as heavy; you'll only feel about 1.26 times heavier. The increased radius of the planet partially offsets the increased mass, saving you from a crushing gravitational pull!